Workshop #2

Data:Posing Answers and Finding Questions

About the Workshop

What is the theme of the workshop?Data-rich activities enliven the mathematics classroom. In this workshop, we'll explore the relationship among situations, data, and graphs, and discuss why we should care about statistics and data in the math curriculum.

Whom do we see? What happens in the videoclips?We'll see students from kindergarten through middle school doing age-appropriate activities with data, ranging from the very beginning - learning to represent quantities with symbols - to the beginnings of statistical inference. We'll see kids use data from the real world and estimation as well as "pure mathematical" data.

What issues does this program address?This workshop addresses a range of issues. For example, how does understanding develop over the years? What role does data play in helping more students succeed in mathematics? How do you decide whether to use "real" or "made-up" data?

What teaching strategy does this program offer?In this workshop, we see teachers using good general principles for using data in the math classroom: using data children collect and generate themselves; using data related to students' lives, either through experience or interest; asking children both to create and to interpret representations of data; and using data to support other areas of the curriculum (e.g., by plotting estimates on a number line, which supports developing number sense and estimation.)

To which NCTM Standards does this workshop relate?This workshop focuses primarily on Standard 11: Statistics and Probability (elementary) or Standard 10: Statistics (middle grades). Naturally, other content strands appear, notably Estimation (Elementary #5, Middle #7). All four "process" standards are well-represented, but you will especially see Standards 2 and 4: Communication and Connections.

Suggested Classroom Activities

Collecting and Displaying Data: SurveysIn order to be intelligent - and skeptical-consumers of data, students need experiences collecting and displaying data themselves. Surveys are natural venues for this. Students design their own surveys and (with teacher approval) administer them. Then they make a display of their data and report on what they found. This can be more or less sophisticated depending on the experience of the students: kindergarteners can make picture plots, surveying their own class, and dictate "one thing you can say from the graph." Eighth-graders can make two-way tables with row and column percentages, list questions you can answer based on the data, list possible causes and alternative hypotheses, and comment on possible bias in the sample.

Collecting and Displaying Data: MeasurementsContinuous data has a different character from typical, categorical, survey data (in which the subject answers questions with a finite number of choices, e.g., yes/no or vanilla/chocolate/strawberry). Displaying and analyzing these data involve point plots, scatter plots (to see trends and relationships), box plots, and using summary statistics such as mean, median, and interquartile range. Again, students can make a display and a write-up. Topics might include: comparing before and after estimates; comparing heights; how far balls bounce or paper airplanes fly; how long you can hold your breath or stand on one foot blindfolded; how many minivans are in the parking lot at the mall; or repeated trials of doing ten free-throws in a row.

Analyzing Published DataGet graphs or tables from anywhere (USA Today is one good choice) and let students analyze them. At first, the point is simply to tell in your own words what the graph or table is saying. Later, students can figure out the context and what point the author is trying to make. Then they can come up with alternative explanations, and alternative representations with the same data.

Peer AssessentAny of these small projects are perfect fodder for peer assessment. If students make posters that they put up around the room, they might also be asked to look at three other posters and comment on them. Consider applying "writers' workshop" commenting guidelines, e.g., students write answers to "What did you like about this poster/presentation?" and "What additional questions do you have?" to help make the situation safe for criticism. In this way, each student gets feedback and has the experience of giving it; all of this supports the goal of fostering communication skills in mathematics.

Suggested Strategies

We'll see teachers using a variety of instructional techniques. During the workshop, pay special attention to how the teachers used cooperative groups and student presentations. How would these work in your own classroom?

Post-Workshop Questions

Students will naturally become more sophisticated thinkers as they grow older. But when, exactly, do some data-oriented ideas become accessible? In particular, when can students begin to make and interpret various kinds of graphs (e.g., point plots, circle graphs, scatter plots, box-and-whisker plots)? When can students begin to reason about differences among groups in their data? When should we expect students to begin using statistical measures such as mean, median, etc., spontaneously? And how does their understanding develop over time?

How do you decide if it's worth the class's time - and your effort - to do an activity that may span more than one period? What do you do afterwards to evaluate for yourself whether it was worth it?

When students give a wrong answer or use faulty reasoning in a presentation, how do you deal with that?

How do you feel about your own skills and experience with data and statistics?

Pre Workshop Assignment for Workshop #3

A "net" is a set of attached squares. How many different nets can you find that can be folded along the lines to make a cube?Each net should have six squares so there's no overlap. Here is one net that works:

In the workshop, you will be asked to describe your nets using numbers from the grid below. The solution we described could be called, "4-7-8-9-10-16."Note that a net can have more than one numerical description. For example, the net described above could also be described as "1-7-8-9-10-13."